Problem 33 - Algebra 1

Question

Simplify the expression. The simplified expression should have no negative exponents. $$ \left(\frac{1}{x}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{x^{5}} \)

Step 1: Write down the expression

Write down the expression as it is \( \left(\frac{1}{x}\right)^{5} \).

Step 2: Apply the exponent rule

In the given expression, x is in the denominator with an implied exponent of -1. This exponent needs to be multiplied with the given exponent which is 5. Hence, use the rule am.n = a^(m*n). So the expression becomes \( x^{-5} \).

Step 3: Simplify the expression

The result obtained in Step 2 is \( x^{-5} \). We need to convert this negative exponent to a positive exponent. The rule to convert negative exponent to positive is a^(-n) = 1/a^n. Applying this rule, the answer simplifies to \( \frac{1}{x^{5}} \).

Key Concepts

Exponent Rules

Understanding exponent rules is essential in simplifying expressions involving exponents. Exponents are a shorthand way of expressing repeated multiplication of the same number. For example, \( x^5 \) means \( x \times x \times x \times x \times x \). There are several key rules when working with exponents:

• Product of Powers: \( a^m \times a^n = a^{m+n} \). This means when you multiply like bases, you add the exponents.
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \). When dividing like bases, you subtract the exponents.
• Power of a Power: \( (a^m)^n = a^{m \times n} \). When raising a power to another power, you multiply the exponents.
• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \). This essentially "flips" the base to a reciprocal to make the exponent positive.

Using these rules helps in converting complex expressions to a more straightforward form. In our exercise, we used the negative exponent rule to convert from a negative exponent to a positive exponent.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves using mathematical rules and operations to make expressions as concise and clear as possible. It's like "cleaning up" the math mess! Let's take a look at how this process typically works:

• Identify like terms or similar bases, allowing for addition, subtraction, or application of exponent rules.
• Apply exponent rules to handle powers and simplify expressions fully. Focus on changing negative exponents to positive if needed.
• Simplify fractions by factoring numerators and denominators if possible, then reduce to simplest form.
• Keep expressions as neat as possible to reduce computational complexity and maintain clarity.

During the exercise, we specifically concentrated on applying exponent rules. By switching \( x^{-5} \) to \( \frac{1}{x^5} \), we simplified the expression fully while keeping terms positive and easy to handle.

Positive Exponents

Working with positive exponents is often simpler than dealing with negative ones. A positive exponent indicates how many times to multiply a number by itself. For example, with \( x^5 \), you multiply 5 \( x \)'s together. Positive exponents generally make calculations clearer and easier to manage.Unlike negative exponents, which represent division or reciprocals, positive exponents mean consistent multiplication. When converting negative exponents to positive exponents, you use the reciprocal of the base. For example:- Given \( x^{-n} \), the positive form would be \( \frac{1}{x^n} \).- This conversion ensures all expressions have positive exponents, which is typically a requirement for simplified expressions.In the task, transforming \( x^{-5} \) into \( \frac{1}{x^5} \) made the mathematical expression much simpler and clearer. This clarity is fundamental for further calculations and when integrating such expressions in more extensive algebraic work.